
In (REF ), \(f_i(X) := \mathbb {E}_{j \sim \mathcal {D}_i} \big [ f_i^j(X) \big ]\), and \(f_i^j(X)\)  is the loss function evaluated at the \(j\) -th observation of the locally stored dataset of machine \(i\) , which follows the distribution \(\mathcal {D}_i.\)We assume the homogeneous data case where \(\mathcal {D}_i = \mathcal {D}\)  for all \(i\) .
The function \(f_i: \mathbb {R}^{d \times d} \rightarrow \mathbb {R}\)  is a (restricted) strongly convex/smooth differentiable function, and \(X \succeq 0\)  is the set of positive semi-definite matrices.We provide theory for the real case; extensions to complex domains can be obtained with complex conversions and Wirtinger derivatives [1]}.
